Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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48 views

How is topology related to physics?

Topology has many occurences in physics like topological insulators, topological quantum computing etc. But what is confusing me is that topology is this mathematical theory that studies the behaviour ...
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Theoretically, how could a wormhole be made? [duplicate]

Concentrating a lot of matter in one place will make a black hole, not a wormhole. A wormhole would change the topology of spacetime. Does General Relativity allow this? I know there are wormhole ...
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Deforming a nematic line defect to a uniform configuration

In Nakahara section 4.9, "Defects in nematic liquid crystals", it is discussed that the order parameter for a nematic should be the real projective plane $\mathbb{R}P^2$, which has ...
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How does the Bloch sphere indicate topology of 2-level $k\cdot p$ effective Hamiltonians?

It is known that the topology of some parameter space of a 2-level system (such as the Brillouin torus) may be found via the Gauss map to the Bloch sphere. The topology is indicated by the number of ...
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What is topological in Kitaev Chain

What is topological in Kitaev Chain? Realspace or the space of Pauli spins or the space of fermions? My Understanding I understand that majorana-zero modes are which are spatially separated, are ...
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Cosmic strings, monopoles and textures

I am a beginner in topology and I want some help in visualizing the following statements: "Strings arise when manifold contains non-contractible loops" "Monopoles arise when manifold ...
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1answer
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Does spacetime topology have importance in physics?

Generally in textbooks they represent spacetime as $(M,\nabla,g,t)$ where $M$ is a Lorentzian manifold,$\nabla$ a torsion-free connection,$g$ a metric and $t$ a time orientation. But they do not talk ...
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Why doesn't the $3d$ gravitino have a quantized “level”?

The action for the $3d$ gravitino is $$S_g=-\int d^3x\bar{\Psi}_{\mu}\gamma^{\mu\lambda\nu}\partial_{\lambda}\Psi_{\nu}$$ Where $\gamma^{\mu\lambda\nu}=-\epsilon^{\mu\lambda\nu}$. This has a striking ...
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Doubt on the need for Topological Manifolds [closed]

I happen to be trying to motivate physically and intuitively the need to use topological spaces. So without using emerging concepts such as differentiable manifolds or extremely formal concepts as ...
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3answers
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Particle Hole Symmetry of BdG Hamiltonians

It is straight-forward to verify that any Hermitian BdG Hamiltonian of the form $$ \mathcal{H} = (c_1^\dagger, c_1, c_2^\dagger, c_2,...) \begin{pmatrix} H_{11} & H_{12} & \cdots \\ H_{21} &...
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Is this a solution of Einstein's equations?

Take infinite space. $\mathbb{R}^3$ Then cut a sphere (a 3-ball) out of it and discard it. You now have $\mathbb{R}^3\backslash B_3$. Now take each point on the surface of the hole and identify it ...
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Is friction required for knots (not hitches or bends) to hold fast to a rope

Using rope with idealized coefficient of surface friction approaching zero, that is in all other ways typical (typically stretchable, compressible, flexible and twistable), is it possible to tighten a ...
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Anti-de Sitter Spacetime Properties

I learned from reading nLab (https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime) that the anti-de Sitter Spacetime of dimension $d$, $AdS_d$, is homeomorphic to $\mathbb{R}^{d-1} \times S^1$. I ...
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Spontaneous discrete symmetry breaking always implies domain walls

I've read several times that if a discrete symmetry is spontaneously broken, then there exist domain walls that interpolate between the different vacua. However, Weinberg says that if the former ...
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Are All Dimensions Infinite and 'Closed'? [closed]

I have formed a thread about loosely how a 'fluid', geometrically flat, N-space, that is 'curved', in infinity'. With flows of fluid, 'space', that can facilitate known forces in that flat space. In ...
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Waves on a topological surface

Are there any formulas for wave motion on a topological surface, like a Mobius strip? If not, is this a valid opportunity for research?
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Gauge cosmic strings and large gauge transformations

I've been going around in circles (hah) about how gauged cosmic strings work (I've been using Preskill's notes for the most part). The global string scenario makes sense to me, since different points ...
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1answer
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What is the meaning of “any real objects exist in open sets”?

I was reading Isham, Chris J. Modern differential geometry for physicists. Vol. 61. World Scientific, 1999. p.52 In the first chapter, he gives mathematical preliminaries that'll be useful for the ...
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What is the magnetic flux through a trefoil knot?

Imagine a closed loop in the shape of a trefoil knot (https://en.wikipedia.org/wiki/Trefoil_knot). How should one calculate the flux through this loop? Normally we define an arbitrary smooth surface, ...
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Sum of topological charges is the Euler characteristic

I have seen many places claiming that the given a collection of topological defects on a 2-dimensional surface, the sum of the topological charges is $2\pi\chi$ (where $\chi$ is the Euler ...
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2answers
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A path for learning General Relativity formally [closed]

I was a physics major a couple years ago and took a few undergrad and grad general relativity classes, got decently good at it, but changed majors and have forgotten most of the stuff. I wanted to ...
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1answer
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Boundary condition of gauge field for finite Euclidean action

I am reading the book "Gauge theory of elementary particle physics" by Cheng & Li chapter 16 and I am confused by some statements. In Euclidean 4D spacetime we have a $SU(2)$ gauge teory ...
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Vacuum manifold of Glashow-Weinberg-Salam & lack of vortex solutions

Update: may have figured it out (below) I've been reading Preskill's notes on vortices, and he's included a couple of exercises at the end of section 1.3. In the first, you must look at the breaking ...
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Verlinde formula requires compact Riemann surface

Assume a Chern-Simons theory on a 3-mfld of the form $\Sigma_g \times S^1$, where $\Sigma_g$ is a Riemann surface of genus $g$. In his paper, Witten shows that one can use the Verlinde formula to get ...
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Weyl point? Weyl node?

I started studying Weyl physics in condensed matters, but I got confusing about the difference between the Weyl point and Weyl node. I understood that when the Weyl points connect continuously, the ...
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Choice of metric/topology on $\mathbb{R}^n$ when we say a manifold is locally homeomorphic to it

I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modelled as a topological manifold (with a bunch of additional structure that's not relevant to this ...
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Why would one want to study the geometry of Lie groups?

Lie groups are commonly used in theoretical physics and mathematical physics. They are useful tools to study simple systems such as the harmonic oscillator. They are also crucial in representation ...
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Expectation value of Wilson loop in Chern-Simons theory

I have read Witten's paper, and I am interested on computing the expectation value of a Wilson loop with a representation $R$ on Chern-Simons theory in $d=3$. I am especially interested in cases for $...
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482 views

What does it mean that the event horizon is a global concept?

In Chapter 6 of Spacetime and Geometry by S. Carroll, he says ‘‘Because the event horizon is a global concept, it might be difficult to actually locate one when you are handed a metric in an ...
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How do you calculate curvature from the density?

We know that this density results in a flat universe. $$\rho_c=\frac{3 H^2}{8 \pi G}$$ And we know that if the universe isn't flat, the density as a proportion of critical density can be expressed as ...
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1answer
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Instantons in 1+1 dimensional Abelian Higgs model

Let's consider the Abelian Higgs model in 1+1 dimensions in Euclidean space-time: $$L_E=\frac{1}{4e^2}F_{\mu\nu}F_{\mu\nu}+D_\mu\phi^\dagger D_\mu\phi+ \frac{e^2}{4}(|\phi|^2-\zeta)^2$$ where $\zeta&...
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Winding number of center symmetries

Suppose we have a center symmetry transformation that acts on the connection as: $$A \mapsto \ ^gA = \Omega^{-1} (A+d)\Omega $$ and satisfies $$ \Omega(t+\beta,x) = h\Omega(t,x)$$ Suppose $W[U]$ ...
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Why is the “instanton map” surjective and do we compactify the space or not?

The following line of reasoning, apart from possible misconceptions in my part, is how instantons are usually (intuitively, at least) introduced: (i) We look for minimum classical action solution for ...
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If the universe is closed, does that also mean time is closed?

Speaking just about space, we say that the universe is either open (topologically $E^3$) or closed (topologically $S^3$). But since a metric connection defines curvature on spacetime and not just ...
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Does positive curvature imply a closed universe?

Topologically speaking, our universe is either open (topologically $E^3$) or closed (topologically $S^3$). Then with time we'd have another factor of $E^1$ and a metric connection would determine the ...
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Different action principle for describing (maybe different) dynamics of string

In the lectures of string theory by David Tong, he gave a simple idea to come up with action of closed string, it's the area of worldsheet since its independent of our coordinate choice (...
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1answer
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Deriving the Gauss-Bonnet term from the Euler class

The Gauss-Bonnet term is just the Euler class of $4D$ manifolds. The Euler class is defined as $$e(\Omega) = \text{Pf}(\Omega)$$ where $\Omega$ is the curvature two-form and $\text{Pf}(\Omega)$ is its ...
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1answer
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What is the Topologically Twisted Index?

I know that one can take a supersymmetric theory defined on $\mathbb{R}^n$ and topologically twist it by redefining the rotation group of the theory into a mixture of the (spacetime) rotation group ...
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What exactly is an Orbifold?

What exactly is an orbifold? I've come across orbifolds on several occasions and I know they are important to string theory, but what is an orbifold? I've seen some very technical mathematical ...
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1answer
107 views

What is a Topological Twist?

I have come across topological twists on numerous occasions but I have never actually seen them explained in an understandable way. So, I was wondering What does it physically mean to topologically ...
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Phase transition in 2+1D QED for $\theta =\pi$

QED in 1+1 dimensions with a massive fermion, called the massive Schwinger model, has a phase transition at $\theta=\pi$ for some finite critical mass $m_c$. Equivalently, one can say that the ...
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Is “Particle in a box” actually a misnomer?

In the usual statement of the Particle in a Box problem, we assume two infinite potential barriers, to hold its wavefunction constrained, so it goes to zero on both ends: But instead of invoking some ...
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Integrating over non-trivial fiber bundles - Chern-Simons Theory

I have been reading Tong's notes on QHE and Gauge Theories, specifically the part about quantizing the Abelian U(1) Chern-Simons level at finite temperature in the presence of a monopole (These ...
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What is the general form of the Friedmann-Robertson-Walker metric in 4D?

I have written in some old notes that the FLRW (also known as FRW) metric can be written as: $$ds^2=dt^2 + a^2 (t) [dr^2 +r^2(d\theta^2 + sin^2\theta d\varphi ^2)] \tag{1}$$ I believe this is its ...
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The generic form of instanton (antiinstanton) in Kahler $\sigma$-model

Suppose we have some compact Riemann surface $\Sigma$ , and scalar field $\phi$, which takes values in some Kahler manifold (target space) $M$. In other words, we have a map: $$ \phi : \Sigma \...
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Inversion symmetry on surface and spin

Let us assume you have a 3D bulk periodic crystal which has inversion symmetry e.g. $r\rightarrow -r$. Assume we are considering spinful operators with $S=1/2$. Now let us imagine cutting a surface ...
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Can a null hypersurface be foliated by spacelike sections?

Let $(M,g)$ be a $d$-dimensional Lorentzian manifold and let $\Sigma \subset M$ be a null hypersurface, which therefore has dimension $(d-1)$. We know that its normal vector $k^\mu$ is null and since ...
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1answer
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Index on a compact manifold

How can the integral of a topological term (like the Nieh-Yan term) on all of a compact manifold be nonzero whereas it's a total derivative and the manifold has no boundary? I assume the manifold can ...
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2answers
156 views

How to know if a Feynman diagram is planar?

A planar diagram is defined as being one of the leading diagrams for $N \to \infty$ (large $N$ expansion), and, as I understand it, it should have the lowest genus when compared to a non-planar ...
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Questions about Kosterlitz–Thouless (KT) transition

Why we extend $\theta$ from $(0,2\pi)$ to $(-\infty, \infty)$? I mean we cannot measure $\theta$ in experiment, can we? Secondly,the feature of vortex solution (at least in KT transition) can be ...

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